ASYMPTOTIC EQUIVALENCE OF LOCALLY STATIONARY PROCESSES AND BIVARIATE GAUSSIAN WHITE NOISE

• 成果类型: Article
• 署名作者: Butucea, Cristina; Meister, Alexander; Rohde, Angelika
• 署名单位: Institut Polytechnique de Paris; ENSAE Paris; University of Rostock; University of Freiburg
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/25-AOS2491
• 发表日期: 2025
• 页码: 879-906
• 关键词: Nonparametric regression; statistical equivalence; DENSITY-ESTIMATION; approximation

摘要

We consider a general class of statistical experiments, in which an ndimensional centered Gaussian random variable is observed and its covariance matrix is the parameter of interest. The covariance matrix is assumed to be well-approximable in a linear space of lower dimension Kn with eigenvalues uniformly bounded away from zero and infinity. We prove asymptotic equivalence of this experiment and a class of Kn-dimensional Gaussian models with informative expectation in Le Cam's sense when n tends to infinity and Kn is allowed to increase moderately in n at a polynomial rate. For this purpose we derive a new localization technique for non-i.i.d. data and a novel high-dimensional Central Limit Law in total variation distance. These results are key ingredients to show asymptotic equivalence between the experiments of locally stationary Gaussian time series and a bivariate Wiener process with the log spectral density as its drift. On this way a novel class of matrices is introduced which generalizes circulant Toeplitz matrices traditionally used for strictly stationary time series.